Computational Tools for Topological Cohochschild Homology 3

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Subject Mathematics h oooyo oH o oler pcr.W s coalgeb computations. a several use produce We to spectra. sequence coalgebra spectral this for on coTHH of s homology coBökstedt a spectral the develop also c parti We cofree In coalgebras. the theory. graded in new theorem this t type study In Hochschild-Kostant-Rosenberg to coalgebras. tools computational for velop (coTHH) homology coHochschild logical Abstract. OPTTOA OL O TOPOLOGICAL FOR TOOLS COMPUTATIONAL ROESILY N TPAI ZIEGENHAGEN STEPHANIE AND SHIPLEY, BROOKE nrcn ok esadSily[8 endater ftopo- of theory a defined [18] Shipley and Hess work, recent In OOHCIDHOMOLOGY COHOCHSCHILD oooia ohcidhmlg,cagba Hochschild– coalgebra, homology, Hochschild Topological 1. Introduction rmr:1D5 61;Scnay 64,55U35, 16E40, Secondary: 16T15; 19D55, Primary: 1 qec ocompute to equence s o differential for ase ua,w rv a prove we cular, i ae ede- we paper his l ont nanal- an to down ils astructure ra a n[,Section [7, in ear eea situation general agba,there oalgebras, iis ethen We limits. ,w iemore give we s, exts. to fcofree of otion ant–Rosenberg lsia result classical topological a e lHochschild al dnie the identifies Hochschild y olersis coalgebras symmetric a ilgraded tial Variations homology c.This ect. iiga giving Hoch- e Kostant– alogue, 2 BOHMANN, GERHARDT, HØGENHAVEN, SHIPLEY, AND ZIEGENHAGEN prove our first computational result, a Hochschild–Kostant–Rosenberg theorem for cofree differential graded coalgebras. A similar result has been obtained by Farinati and Solotar in [13] and [14] in the ungraded setting. Theorem 1.1. Let X be a nonnegatively graded cochain complex over a field k. Let Sc(X) be the cofree coaugmented cocommutative coassociative conilpotent coalgebra over k cogenerated by X. Then there is a quasi-isomorphism of differential graded k-modules c coTHH(Sc(X)) ≃ ΩS (X)|k. where the right hand side is an explicit differential graded k-module defined in Sec- tion 3. c The precise definition of ΩS (X)|k depends on the characteristic of k. Let U(Sc(X)) denote the underlying differential graded k-modules of the cofree coalgebra Sc(X). If char(k) 6= 2, we can compute coTHH(Sc(X)) as U(Sc(X))⊗U(Sc(Σ−1X)), while in characteristic 2, we have coTHH(Sc(X)) =∼ U(Sc(X)) ⊗ Λ(Σ−1X), where Λ denotes exterior powers. Definitions of all of the terms here appear in Sections 2 and 3 and this theorem is proved as Theorem 3.15. We then develop calculational tools to study coTHH(C) for a coalgebra spectrum C in a symmetric monoidal model category of spectra; see Section 4. Recall that for topological Hochschild homology, the Bökstedt spectral sequence is an essential computational tool. For a field k and a ring spectrum R, the Bökstedt spectral sequence is of the form 2 E∗,∗ = HH∗(H∗(R; k)) ⇒ H∗(THH(R); k). This spectral sequence arises from the skeletal filtration of the simplicial spectrum THH(R)•. Analogously, for a coalgebra spectrum C, we consider the Bousfield– Kan spectral sequence arising from the cosimplicial spectrum coTHH•(C). We call this the coBökstedt spectral sequence. We identify the E2-term in this spectral sequence and see that, as in the THH case, the E2-term is given by a classical algebraic invariant: Theorem 1.2. Let k be a field. Let C be a coalgebra spectrum that is cofibrant as an underlying spectrum. The Bousfield–Kan spectral sequence for coTHH(C) gives a coBökstedt spectral sequence with E2-page s,t k E2 = coHHs,t(H∗(C; k)), that abuts to Ht−s(coTHH(C); k). As one would expect with a Bousfield–Kan spectral sequence, the coBökstedt spectral sequence does not always converge. However, we identify conditions under which it converges completely. In particular, if the coalgebra spectrum C is a ∞ suspension spectrum of a simply connected space X, denoted Σ+ X, the coBökstedt ∞ spectral sequence converges completely to H∗(coTHH(Σ+ X); k) if a Mittag-Leffler condition is satisfied. Further, we prove that for C connected and cocommutative this is a spectral sequence of coalgebras. Using this additional algebraic structure we prove several computational results, including the following. The divided power coalgebra Γk[−] and the exterior coalgebra Λk(−) are introduced in detail in Section 5. Note that if X is a graded k-vector space concentrated in even degrees and with basis (xi)i∈N, COMPUTATIONAL TOOLS FOR TOPOLOGICAL COHOCHSCHILD HOMOLOGY 3

c Sc(X)|k there are identiﬁcations S (X) =∼ Γk[x1, x2,... ] and Ω =∼ U(Γk[x1, x2,... ])⊗ U(Λk[z1,z2,... ]). We improve Theorem 1.1 in Proposition 5.1 by showing that under the above conditions on X there is indeed an isomorphism of coalgebras c coHH(S (X)) =∼ Γk[x1, x2,... ]) ⊗ Λk[z1,z2,... ]. Hence Theorem 1.2 and the coalgebra structure on the spectral sequence yield the following result. Theorem 1.3. Let C be a cocommutative coassociative coalgebra spectrum that is coﬁbrant as an underlying spectrum, and whose homology coalgebra is

H∗(C; k)=Γk[x1, x2,... ], where the xi are cogenerators in nonnegative even degrees, and there are only ﬁnitely many cogenerators in each degree. Then the coB¨okstedt spectral sequence for C collapses at E2, and

E2 =∼ E∞ =∼ Γk[x1, x2,... ] ⊗ Λk(z1,z2,... ), with xi in degree (0, deg(xi)) and zi in degree (1, deg(xi)). This theorem should thus be thought of as a computational analogue of the Hochschild–Kostant–Rosenberg Theorem at the level of homology. These computational results apply to determine the homology of coTHH of many suspension spectra, which have a coalgebra structure induced by the diagonal map ∞ on spaces. For simply connected X, topological coHochschild homology of Σ+ X ∞ coincides with topological Hochschild homology of Σ+ ΩX, and thus is closely re- lated to algebraic K-theory, free loop spaces, and string topology. Our results give a new way of approaching these invariants, as we briefly recall in Section 5. The paper is organized as follows. In Section 2, we define coTHH for a coalgebra in any model category endowed with a symmetric monoidal structure as the homotopy limit of the cosimplicial object given by the cyclic cobar construction coTHH•. We then give conditions for when this homotopy limit can be calculated efficiently and explain a variety of examples. In Section 3, we develop cofree coalgebra functors. These are not simply dual to free algebra functors, and we are explicit about the conditions on a symmetric monoidal category that are necessary to make these functors well behaved. In this context an identification similar to the Hochschild–Kostant–Rosenberg Theorem holds on the cosimplicial level. We also prove a dg-version of the Hochschild– Kostant–Rosenberg theorem. In Section 4 we define and study the coBökstedt spectral sequence for a coalgebra spectrum. In particular, we analyze the convergence of the spectral sequence, and prove that it is a spectral sequence of coalgebras. In Section 5 we use the coBökstedt spectral sequence to make computations of the homology of coTHH(C) for certain coalgebra spectra C. We exploit the coalgebra structure in the coBökstedt spectral sequence to prove that the spectral sequence collapses at E2 in particular situations and give several specific examples. 1.1. Acknowledgements. The authors express their gratitude to the organizers of the Women in Topology II Workshop and the Banff International Research Station, where much of the research presented in this article was carried out. We also thank Kathryn Hess, Ben Antieau, and Paul Goerss for several helpful conversations. The second author was supported by the National Science Foundation CAREER 4 BOHMANN, GERHARDT, HØGENHAVEN, SHIPLEY, AND ZIEGENHAGEN

Grant, DMS-1149408. The third author was supported by the DNRF Niels Bohr Professorship of Lars Hesselholt. The fourth author was supported by the National Science Foundation, DMS-140648. The participation of the ﬁrst author in the workshop was partially supported by the AWM’s ADVANCE grant.

2. coHochschild homology: Definitions and examples Let (D, ⊗, 1) be a symmetric monoidal category. We will suppress the associa- tivity and unitality isomorphisms that are part of this structure in our notation. The coHochschild homology of a coalgebra in D is defined as the homotopy limit of a cosimplicial object in D. In this section, we define coHochschild homology and then discuss a variety of algebraic examples. Definition 2.1. A coalgebra (C, △) in D consists of an object C in D together with a morphism △: C → C ⊗ C, called comultiplication, which is coassociative, i.e. satisfies (C ⊗△)△ = (△⊗ C)△. Here we use C to denote the identity morphism on the object C ∈ D. A coalgebra is called counital if it admits a counit morphism ǫ: C → 1 such that (ǫ ⊗ C)△ = C = (C ⊗ ǫ)△. Finally, a counital coalgebra C is called coaugmented if there furthermore is a coaugmentation morphism η : 1 → C satisfying the identities △η = η ⊗ η and ǫη =1.

We denote the category of coaugmented coalgebras by CoalgD. The coalgebra C is called cocommutative if

τC,C △ = △, with τC,D : C ⊗ D → D ⊗ C the symmetry isomorphism. The category of cocommutative coaugmented coalgebras in D is denoted by comCoalgD. We often do not distinguish between a coalgebra and its underlying object. Given a counital coalgebra, we can form the associated cosimplicial coHochschild object: Deﬁnition 2.2. Let (C, △,ǫ) be a counital coalgebra in D. We deﬁne a cosimplicial object coTHH•(C) in D by setting coTHHn(C)= C⊗n+1 with cofaces ⊗i n−i ⊗n+1 ⊗n+2 C ⊗△⊗ C , 0 ≤ i ≤ n, δi : C → C , δi = ⊗n (τC,C⊗n+1 (△⊗ C ), i = n +1, and codegeneracies ⊗n+2 ⊗n+1 ⊗i+1 ⊗n−i σi : C → C , σi = C ⊗ ǫ ⊗ C for 0 ≤ i ≤ n. Note that if C is cocommutative, △ is a coalgebra morphism and hence coTHH•(C) is a cosimplicial counital coalgebra in D. Recall that if D is a model category, by [19, 16.7.11] the category D∆ of cosimplicial objects in D is a Reedy framed diagram category, since ∆ is a Reedy category. In particular, simplicial frames exist in D. Hence we can form the homotopy limit COMPUTATIONAL TOOLS FOR TOPOLOGICAL COHOCHSCHILD HOMOLOGY 5

• ∆ holim∆X for X ∈ D . More precisely, we use the model given by Hirschhorn [19, Chapters 16 and 19] and in particular apply [19, 19.8.7], so that holim∆(X•) can be computed via totalization as

holim∆X• = Tot Y• where Y• is a Reedy fibrant replacement of X•. Note that in order to apply [19, 19.8.7], it suffices to have a Reedy framed diagram category structure. Definition 2.3. Let D be a model category and a symmetric monoidal category. Let (C, △, 1) be a counital coalgebra in D. Then the coHochschild homology of C is defined by • coTHH(C) = holim∆coTHH (C). We chose to denote coHochschild homology in a category D as above by coTHH instead of coHH to emphasize that the construction relies on the existence of a model structure on D. However, in several of the examples that follow, the category D does not come from topological spaces or spectra. For example we will see in 2.8 and 2.12 that coTHH coincides with the origi- nal definition of coHochschild homology for k-modules and more generally cochain coalgebras (see [11], [17]). We will use the standard notation coHH(C) for the co- Hochschild homology of a coalgebra C in the category of graded k-modules in later sections. Remark 2.4. Let D be a symmetric monoidal model category. Let f : C → D be a morphism of counital coalgebras and assume that f is a weak equivalence and C and D are cofibrant in D. Then f induces a degreewise weak equivalence of cosimplicial objects coTHH•(C) → coTHH•(D). Hence the corresponding Reedy fibrant replacements are also weakly equivalent and f induces a weak equivalence coTHH(C) → coTHH(D) according to [19, 19.4.2(2)]. A central question raised by the definition of coHochschild homology is that of understanding the homotopy limit over ∆. In the rest of this section, we discuss conditions under which one can efficiently compute it, including an assortment of algebraic examples which we discuss further in Section 3. • • For a cosimpicial object X in D let Mn(X ) be its nth matching object given by • a Mn(X ) = lim X . α∈∆([n],[a]), α surjective, a6=n • n+1 • The matching object Mn(X ) is traditionally also denoted by M (X ), but we stick to the conventions in [19, 15.2]. If D is a model category, X• is called Reedy fibrant if for every n, the morphism n • σ : X → Mn(X ) n a is a fibration, where σ is induced by the collection of maps α∗ : X → X for α ∈ ∆([n], [a]). • • If D is a simplicial model category and X is Reedy fibrant, holim∆X is weakly equivalent to the totalization of X• defined via the simplicial structure; see [19, Theorem 18.7.4]. In this case the totalization is given by the equalizer

n a Tot(X•)=eq (Xn)∆ ⇒ (Xb)∆ , n ! Y≥0 α∈∆([Ya],[b]) 6 BOHMANN, GERHARDT, HØGENHAVEN, SHIPLEY, AND ZIEGENHAGEN where DK denotes the cotensor of an object D of D and a simplicial set K and where the two maps are induced by the cosimplicial structures of X• and ∆•. If the category D is not simplicial, one has to work with simplicial frames as in [19, Chapter 19]. In the dual case, if A is a monoid in a symmetric monoidal model category, one considers the cyclic bar construction associated to A, i.e. the simplicial object whose definition is entirely dual to the definition of coTHH•(C). This simplicial object is Reedy cofibrant under mild conditions; see for example [34, Example 23.8]. But the arguments proving Reedy cofibrancy of the simplicial Hochschild object do not dualize to coTHH•(C): It is neither reasonable to expect the monoidal product to commute with pullbacks, nor to expect that pullbacks and the monoidal product satisfy an analogue of the pushout-product axiom. Nonetheless, in certain cases coTHH•(C) can be seen to be Reedy fibrant. In particular Reedy fibrancy is simple when the symmetric monoidal structure is given by the cartesian product. Lemma 2.5. Let D be any category and let C be a coalgebra in (D, ×, ∗). Then n • σ : coTHH (C) → Mn(coTHH (C)) is an isomorphism for n ≥ 2. Hence if D is a model category, then coTHH•(C) is Reedy fibrant if C itself is fibrant. Proof. For surjective α ∈ ∆([n], [n − 1]), let • n−1 ×n pα : Mn(coTHH (C)) → coTHH (C)= C be the canonical projection to the copy of coTHHn−1(C) corresponding to α. Then an inverse to σ is given by

×n ×n ×n ×n−1 • (pα)α ×n ×n ((C ) ,C ×(∗) ) ×n n+1 Mn(coTHH (C)) −−−−→ (C ) −−−−−−−−−−−−−−−−−→ (C )

(C×∗n−1)×···×(∗n−1×C)×(∗n−1×C) −−−−−−−−−−−−−−−−−−−−−−−−→ C×n+1.

1 n • i i i Intuitively, this map sends (c ,...,c ) ∈ Mn(coTHH (C)) with c = (c1,...,cn) ∈ ×n 1 n 1 C to (c1,...,cn,cn). The Reedy fibration conditions for n =1, 0 follow from the fibrancy of C. In several algebraic examples, degreewise surjections are fibrations and the following lemma will imply Reedy fibrancy. Lemma 2.6. Let C be a counital coalgebra in the category of graded k-modules for a commutative ring k. If C is coaugmented, or if there is a k-module map η : k → C such that ǫη = C, then the matching maps n • σ : coTHH (C) → Mn(coTHH (C)) are surjective. In the rather technical proof we construct an explicit preimage for any element • in Mn(coTHH (C)). We defer the proof to Appendix A. Note that if k is a field, then by choosing a basis for C over k compatibly with the counit, we can always construct the necessary map η. Thus coTHH of any counital coalgebra over a field is Reedy fibrant. We now collect a couple of examples of what coTHH is in various categories. COMPUTATIONAL TOOLS FOR TOPOLOGICAL COHOCHSCHILD HOMOLOGY 7

Example 2.7. Let D = (ckmod, ⊗, k) be the symmetric monoidal category of cosimplicial k-modules for a ﬁeld k. There is a simplicial model structure on D; see [15, II.5] for an exposition. The simplicial cotensor of a cosimplicial k-module M • and a simplicial set K• is given by K n n (M ) = Set(Kn,M ) n with α ∈ ∆([n], [m]) mapping f : Kn → M to

α f α∗ Km −→ Kn −→ Mn −−→ Mm, see [16, II.2.8.4]. As in the simplicial case (see [16, III.2.11]), one can check that the ﬁbrations are precisely the degreewise surjective maps. Hence by Lemma 2.6 coTHH(C) can be computed as the totalization, that is, as the diagonal of the bicosimplicial k-module coTHH•(C). Even though D is not a monoidal model category, the tensor product of weak equivalences is again a weak equivalence, and hence a weak equivalence of counital coalgebras induces a weak equivalence on coTHH. Recall that the dual Dold–Kan correspondence is an equivalence of categories N ∗ : ckmod ⇆ dgmod≥0(k) :Γ• between cosimplicial k-modules and nonnegatively graded cochain complexes over k. Given M • ∈ ckmod, the normalized cochain complex N ∗(M •) is deﬁned by

a−1 a • a a−1 N (M )= ker(σi : M → M ) i=0 \ for a> 0 and N 0(M •)= M 0, with diﬀerential d: N a(M •) → N a+1(M •) given by

a+1 i (−1) δi. i=0 X Example 2.8. Let k be a field. Let D be the symmetric monoidal category D = (dgmod≥0(k), ⊗, k) of nonnegatively graded cochain complexes over k. The dual Dold–Kan correspondence yields that D is a simplicial model category, with weak equivalences the quasi-isomorphisms and with fibrations the degreewise surjections. Hence by Lemma 2.6 we have that coTHH•(C) is Reedy fibrant for any counital coalgebra C in D. The totalization of a cosimplicial cochain complex M • is given by Tot(M •) =∼ N ∗(Tot(Γ(M •))) = N ∗(diag(Γ(M •))). h Recall that given a double cochain complex B∗,∗ with differentials d : B∗,∗ → v B∗+1,∗ and d : B∗,∗ → B∗,∗+1, we can form the total cochain complex Totdgmod(B) given by

Totdgmod(B)n = Bp,q. p+q=n Y The projection to the component Bs,n+1−s of the diﬀerential applied to (bp ∈ h p v Bp,n−p)p is given by d (bs−1,n−s+1) + (−1) d (bs,n−s). A similar construction can be carried out for double chain complexes. As in the bisimplicial case, we can turn a bicosimpicial k-module X•,• into a double complex by applying the normalized cochain functor to both cosimplicial 8 BOHMANN, GERHARDT, HØGENHAVEN, SHIPLEY, AND ZIEGENHAGEN directions. The homotopy groups of diagX•,• are the homology of the associated total complex, hence

∗ • coTHH(C) = Totdgmod N (coTHH (C)) for any counital coalgebra in D. Hence our notion of coHochschild homology coincides with the classical one as in [11] and [17]. Again weak equivalences of counital coalgebras induce weak equivalences on coTHH.

Example 2.9. The category D = (sSet, ×, ∗) is a simplicial symmetric monoidal model category with its classical model structure. Every object is cofibrant, and by Lemma 2.5, coTHH(C) can be computed as the totalization of the cosimplicial simplicial set coTHH•(C) when C is fibrant. Example 2.10. Let D = (skmod, ⊗, k) be the category of simplicial k-modules over a field k. Every object is cofibrant in D. Let C be a counital coalgebra. Since every morphism that is degreewise surjective is a fibration (see [16, III.2.10]), Lemma 2.6 yields that coTHH•(C) is Reedy fibrant, and coTHH(C) is given by the usual totalization of cosimplicial simplicial k-modules.

Example 2.11. The category D = (dgmod≥0(k), ⊗, k) of nonnegatively graded chain complexes over a ring k with the projective model structure is both a monoidal model category as well as a simplicial model category via the Dold–Kan equivalence (N, Γ) with (skmod, ⊗, k). If we work over a field, every object is cofibrant, and for any counital coalgebra C in D Lemma 2.6 yields that coTHH•(C) is Reedy fibrant. However, a word of warning is in order. The definition of coHochschild homology given in Definition 2.3 does not coincide with the usual notion of coHochschild homology of a differential graded coalgebra as for example found in [11]: Usu- ally, coHochschild homology of a coalgebra C in k-modules or differential graded k-modules is given by applying the normalized cochain complex functor N ∗ to • ∗ • coTHH (C) and forming the total complex Totdgmod(N (coTHH (C))) of the double chain complex resulting from interpreting N a(coTHH•(C)) as living in chain complex degree −a. But for any cosimplicial nonnegatively graded chain complex M •,

• • Tot(M ) =∼ N∗(Tot(Γ(M ))), where the second Tot is the totalization of the simplicial cosimplicial k-module Γ(M •). As explained in [15, III.1.1.13], we have an identity

• ∗ • N∗(Tot(Γ(M ))) = τ∗(Totdgmod(N (M ))), where τ∗(C) of an unbounded chain complex C is given by

Cn, n> 0, τ∗(C)n = ker(C0 → C−1), n =0, 0,n< 0.

Hence if C is concentrated in degree zero, so is coTHH(C), in contrast to the usual deﬁnition as for example found in [11] and [17]. Nonetheless, if we assume that C0 = 0, the above identiﬁcation of the totalization yields that the two notions of coHochschild homology coincide. COMPUTATIONAL TOOLS FOR TOPOLOGICAL COHOCHSCHILD HOMOLOGY 9

Example 2.12. The category D = (dgmod, ⊗, k) of unbounded chain complexes over a commutative ring k is a symmetric monoidal model category with weak equivalences the quasi-isomorphisms and fibrations the degreewise surjective morphisms; see [21, 4.2.13]. Let C be a cofibrant chain complex. A simplicial frame of C is given by the simplicial chain complex fr(C) which is given in simplicial degree m by the internal m • hom object dgmod(N∗(∆ ), C) (see proof of [21, 5.6.10]). Again, coTHH (C) is • Reedy fibrant. This yields that the homotopy limit holim∆coTHH (C) coincides with the total complex of the double complex N ∗(coTHH•(C)). Again this coincides with the definitions of coHochschild homology in [11] and [17]. Example 2.13. Both the category of symmetric spectra as well as the category of S-modules are simplicial symmetric monoidal model categories. Sections 4 and 5 discuss coTHH for coalgebra spectra.

3. coHochschild homology of cofree coalgebras in symmetric monoidal categories We next turn to calculations of coHochschild homology of cofree coalgebras in symmetric monoidal categories. The first step is to specify what we mean by a “cofree” coalgebra in our symmetric monoidal category D. Intuitively, such a coalgebra should be given by a right adjoint to the forgetful functor from the category of coalgebras in D to D itself. The constructions are the usual ones and can be found for example in [28, II.3.7] in the algebraic case, but we specify conditions under which we can guarantee the existence of this adjoint. For an operadic background to these conditions we refer the reader to [9]. Let D be complete and cocomplete and admit a zero object 0. Given a coalgebra (C, △) in a symmetric monoidal category D and n ≥ 0, we denote by △n : C → C⊗n+1 the iterated comultiplication defined inductively by △0 = C, △n+1 = (△⊗ C⊗n)△n. Note that if (C, △,ǫ,η) is a coaugmented coalgebra, the cokernel of the coaugmentation, coker η, is a coalgebra. Definition 3.1. We call a (non-coaugmented) coalgebra C conilpotent if the morphism n ⊗n+1 (△ )n≥0 : C → C , n Y≥0 that is, the morphism given by △n after projection to C⊗n+1, factors through ⊗n+1 n≥0 C via the canonical morphism

L C⊗n+1 → C⊗n+1. n n M≥0 Y≥0 We say that a coaugmented coalgebra C is conilpotent if coker η is a conilpotent coalgebra. We denote the full subcategory of CoalgD consisting of coaugmented conilpotent conil coalgebras by CoalgD . Similarly, the category of cocommutative conilpotent conil coaugmented coalgebras is denoted by comCoalgD . 10 BOHMANN, GERHARDT, HØGENHAVEN, SHIPLEY, AND ZIEGENHAGEN

Definition 3.2. The symmetric monoidal category (D, ⊗, 1) is called cofree-friendly if it is complete, cocomplete, admits a zero object, and the following additional conditions hold: (1) For all objects D in D, the functor D ⊗− preserves colimits. (2) Finite sums and finite products are naturally isomorphic, that is for every finite set J and all objects Dj, j ∈ J, the morphism

Dj → Dj j J j J M∈ Y∈ induced by the identity morphism on each Dj is an isomorphism. Proposition 3.3. If D is cofree-friendly, the functor conil coker : CoalgD → D, (C, △,ǫ,η) 7→ coker η admits a right adjoint T c, which we call the cofree coaugmented conilpotent coal- c ⊗n gebra functor. The underlying object of T (X) is given by n≥0 X . The comultiplication △ is deﬁned via deconcatenation, i.e. by L

⊗n L (X )a+b=n ∼ X⊗n −−−−−−−−−−−−→n≥0 X⊗n −→= X⊗n n≥0 n≥0 a,b≥0 n≥0 a,b≥0 M M a+Yb=n M a+Mb=n ∼ =∼ X⊗a ⊗ X⊗b −→= X⊗a ⊗ X⊗b, a,b a b M≥0 M≥0 M≥0 where the isomorphism on the first line is provided by condition (2) of Definition 3.2. The counit morphism π : T c(X) → X is the identity on X and the zero morphism on the other summands. Example 3.4. The categories of chain or cochain complexes discussed in Examples 2.8, 2.11 and 2.12 are cofree-friendly, and the cofree coalgebra T c(X) generated by an object in any of these categories is the usual tensor coalgebra. Similarly, the categories of simplicial or cosimplicial k-modules discussed in Ex- amples 2.7 and 2.10 are cofree-friendly, and the cofree coalgebra generated by X is obtained by applying the tensor coalgebra functor in each simplicial or cosimplicial degree. Example 3.5. The categories of simplicial sets, of pointed simplicial sets and of spectra are not cofree-friendly, since finite coproducts and finite products are not isomorphic in any of these categories. A more abstract way of thinking about this is given by observing that in these categories coaugmented coalgebras can not be described as coalgebras over a cooperad in the usual way: While the category of symmetric sequences in these categories is monoidal with respect to the plethysm that gives rise to the notion of operads, the plethysm governing cooperad structures does not define a monoidal product. This is discussed in detail by Ching [9]; see in particular Remark 2.10, Remark 2.20 and Remark 2.21. Let G be a group with identity element e. Recall that a G-action on an object X in D consists of morphisms

φg : X → X, g ∈ G, COMPUTATIONAL TOOLS FOR TOPOLOGICAL COHOCHSCHILD HOMOLOGY 11

G such that φgφh = φgh and φe = X. The fixed points X of this actions are given by the equalizer (X)g∈G G X = eq(X −−−−−⇒ X). (φg )g∈G g G Y∈ Definition 3.6. Recall that for any object X of D the symmetric group Σn acts on X⊗n by permuting the factors. We call D permutation-friendly if the morphism (X⊗a)Σa ⊗ (X⊗b)Σb → (X⊗a+b)Σa×Σb , induced by the morphisms (X⊗a)Σa → X⊗a and (X⊗b)Σb → X⊗b, is an isomorphism for all X in D and all a,b ≥ 0. Proposition 3.7. If D is cofree-friendly and permutation-friendly, the forgetful functor conil coker: comCoalgD → D, (C, △,ǫ,η) 7→ coker η admits a right adjoint Sc, which is called the cofree cocommutative conilpotent coaugmented coalgebra functor. For a given object X of D, define Sc(X) = ⊗n Σn c n≥0(X ) . The comultiplication △ on S (X) is defined by L ((X⊗n)Σn ) L n≥0 a+b=n =∼ (X⊗n)Σn −−−−−−−−−−−−−−−→ (X⊗n)Σn −→ (X⊗n)Σn n≥0 n≥0 a,b≥0, n≥0 a,b≥0, M M a+Yb=n M a+Mb=n

L resa,b a,b≥0 =∼ (X⊗a ⊗ X⊗b)Σa+b −−−−−−−→ (X⊗a ⊗ X⊗b)Σa×Σb a,b a,b M≥0 M≥0 ∼ ∼ −→= (X⊗a)Σa ⊗ (X⊗b)Σb −→= (X⊗a)Σa ⊗ (X⊗b)Σb . a,b a b M≥0 M≥0 M≥0 The isomorphism on the first line again exists because of condition (2) of Definition 3.2 and the isomorphism at the beginning of the last line is derived from Definition 3.6. The map resa,b is the map

⊗a ⊗b Σa+b ⊗a ⊗b Σa×Σb resa,b : (X ⊗ X ) → (X ⊗ X ) c induced by the inclusion Σa × Σb → Σa+b. The counit morphism π : S (X) → X is given by the identity on X and by the zero morphism on the other summands. Example 3.8. If k is a field, all the categories discussed in Example 3.4 are permutation friendly in addition to being cofree friendly. This means that the categories of nonnegatively graded (co)chain complexes and unbounded chain complexes of k-modules and the categories of simplicial and cosimplicial k-modules all admit well-defined cofree cocommutative coalgebra functors. The permutation friendliness of each of these categories can be deduced from the permutation friendliness of the category of graded k-modules. A proof for graded k-modules concentrated in even degrees or for k a field of characteristic 2 can be found in Bourbaki [2, p. IV.49]. If the characteristic of k is different from 2 and M is concentrated in odd degrees, (M ⊗n)Σn and (M ⊗a ⊗M ⊗b)Σa×Σb can be described explicitly in terms of a basis of M. For an arbitrary graded module M, the claim follows from writing M as the direct sum of its even and its odd degree part. More concretely, if M admits a countable basis mi, i ≥ 1, and the mi are even degree elements, or if the characteristic of k is 2, we identify Sc(M) with the Hopf 12 BOHMANN, GERHARDT, HØGENHAVEN, SHIPLEY, AND ZIEGENHAGEN algebra Γk[m1,m2,... ]. As an algebra, Γk[m1,m2,... ] is the divided power algebra generated by M. The isomorphism to Sc(M) is given by identifying the element γ (m ) ··· γ (m ) ∈ Γ [m ,m ,... ] with the element σ.(m⊗j1 ⊗ j1 i1 jn in k 1 2 σ∈sh(j1,...,jn) i1 jn c ··· ⊗ m ) ∈ S (M), where σ ranges over the set of (j1,...,jn)-shuffles. The in P coproduct is given on multiplicative generators by

△(γj (mi)) = γa(mi) ⊗ γb(mi). a b j +X= If M is concentrated in odd degrees and the characteristic of k is different from 2, c we can identify S (M) with the Hopf algebra Λk(m1,m2,... ). This is the exterior algebra generated by M, and we identify mi1 ··· min with σ∈Σn sgn(σ)σ.(m1 ⊗ ···⊗ mn). The coproduct is given by P △(mi)=1 ⊗ mi + mi ⊗ 1. If k is not a field, the cofree cocommutative conilpotent coalgebra cogenerated by a k-module M still exists: It is the largest cocommutative subcoalgebra of T c(M). Now that we have established what we mean by cofree coalgebras in D, we turn to an analysis of coHochschild homology for these coalgebras. These computations should be thought of as a simple case of a dual Hochschild–Kostant–Rosenberg theorem for coalgebras. Let (C, △C ,ǫC , ηC ) and (D, △D,ǫD, ηD) be cocommutative coaugmented coalgebras. Recall that their product in comCoalgD exists and is given by C ⊗ D with comultiplication (C ⊗ τC,D ⊗ D)(△C ⊗△D), counit ǫC ⊗ ǫD and coaugmentation ηC ⊗ ηD. Here τC,D : C ⊗ D → D ⊗ C is the symmetry isomorphism switching C and D. The projections C ⊗ D → C and C ⊗ D → D are given by C ⊗ ǫD and ǫD ⊗ C. Since Sc is a right adjoint, we have the following property: Lemma 3.9. If D is cofree-friendly, permutation-friendly and cocomplete, there is a natural isomorphism Sc(X × Y ) → Sc(X) ⊗ Sc(Y ) c c c in comCoalgD. It is given by the morphisms S (X ×Y ) → S (X) and S (X ×Y ) → Sc(Y ) which are induced by the projections X × Y → X and X × Y → Y . Every category C that admits finite products gives rise to the symmetric monoidal category (C, ×, ∗), where ∗ is the terminal object. Every object X in C is then a counital cocommutative coalgebra with respect to this monoidal structure: the comultiplication X → X × X is the diagonal, the counit X →∗ is the map to the terminal object. We indicate the monoidal structure we use to form coTHH with a • subscript, so that if C is a coalgebra with respect to ⊗, we write coTHH⊗(C). Proposition 3.10. Let D be cofree-friendly and permutation-friendly. Then for any object X in D • c ∼ c • coTHH⊗(S (X)) = S (coTHH×(X)). Proof. Applying Lemma 3.9 yields an isomorphism in each cosimplicial degree n: n c c ⊗n+1 ∼ c ×n+1 c n coTHH⊗(S (X)) = S (X) = S (X )= S (coTHH×(X)). COMPUTATIONAL TOOLS FOR TOPOLOGICAL COHOCHSCHILD HOMOLOGY 13

To check that the cosimplicial structures agree, it suﬃces to show that the comultiplications and counits on both sides agree. The diagonal X → X × X induces a morphism of coalgebras Sc(X) → Sc(X × X). Since Sc(X) is cocommutative, the comultiplication c c c ∼ c △Sc(X) : S (X) → S (X) ⊗ S (X) = S (X × X) is a morphism of coalgebras as well. These maps agree after projection to X × X. A similar argument yields that X → 0 induces the counit of Sc(X). The following theorem is a consequence of this proposition. This is the best result we obtain about the coHochschild homology of cofree coalgebras without making further assumptions on our symmetric monoidal category D. Theorem 3.11. If D is a cofree- and permutation-friendly model category, Propo- sition 3.10 implies that c c • coTHH⊗(S (X)) ∼ holim∆(S (coTHH×(X))). • If D is a simplicial model category, there is an easy description of Tot(coTHH×(X)) as a free loop object. Proposition 3.12 (see e.g. [24]). Let D be a simplicial model category and X a ﬁbrant object of D. Then • S1 holim∆(coTHH×(X)) ∼ X .

This result identifies coTHH×(X) as a sort of “free loop space” on X. This is precisely the case in the category of spaces, as in Example 4.4. Example 3.13. For the category of cosimplicial k-modules we can choose the diagonal as a model for the homotopy limit over ∆. Since the cofree coalgebra cogenerated by a cosimplicial k-module is given by applying Sc degreewise, we actually obtain an isomorphism of coalgebras c c S1 coTHH⊗(S (X)) =∼ S (X ). In the case where Sc is a right Quillen functor, we obtain the following corollary to Theorem 3.11 and Proposition 3.12. Corollary 3.14. Let D be a simplicial and a monoidal model category. Let X in c D be fibrant. If there is a model structure on comCoalgD such that S is a right Quillen functor, then c c S1 coTHH⊗(S (X)) =∼ RS (X ). In general, however, Sc is not a right Quillen functor. Up to a shift in degrees which is dual to the degree shift in the free commutative algebra functor, the most prominent counterexample is the category of chain complexes over a ring k of positive characteristic: The chain complex D2 consists of one copy of k in degrees 1 and 2, with differential the identity. This complex is fibrant and acyclic, but Sc(D2) is not acyclic. This same counterexample shows Sc is not right Quillen either for unbounded chain complexes or nonnegatively graded cochain complexes. Nevertheless, a Hochschild–Kostant–Rosenberg type theorem for cofree coalgebras still holds for coHochschild homology in the category of nonnegatively graded cochain complexes dgmod≥0 over a field. We denote the underlying differential 14 BOHMANN, GERHARDT, HØGENHAVEN, SHIPLEY, AND ZIEGENHAGEN graded k-module of Sc(X) by U(Sc(X)). The desuspension Σ−1X of a cochain complex X is the cochain complex with (Σ−1X)n = Xn−1. Define the graded c k-module ΩS (X)|k by c c −1 Sc(X)|k U(S (X)) ⊗ U(S (Σ X)), if char(k) 6=2, Ω = c −1 (U(S (X)) ⊗ Λ(Σ X), if char(k)=2, with Λ(M) denoting the exterior powers of the cochain complex M. We will com- c pare ΩS (X)|k with the notion of Kähler codifferentials as defined in [13] in Remark 3.19. Theorem 3.15. Let k be a field. Then for X in dgmod≥0 there is a quasi- isomorphism c Sc(X)|k coTHH⊗(S (X)) → Ω . Note that this is an identification of differential graded k-modules, not of coal- c gebras. We determine the corresponding coalgebra structure on coTHH⊗(S (X)) in certain cases in Proposition 5.1. This result corresponds to the result of the Hochschild–Kostant–Rosenberg theorem applied to a free symmetric algebra S(X) generated by a chain complex X. See, for example, [23, Theorems 3.2.2 and 5.4.6]. For a discussion of a coalgebra analogue of Kähler differentials we refer the reader to [13]. The proof is dual to the proof of the corresponding results for Hochschild homology. We follow the line of proof given by Loday [23, Theorem 3.2.2]. We begin by proving a couple of lemmas. First we identify coTHH×(X): Lemma 3.16. For a cochain complex X over a field k, ∗ • ∼ −1 Totdgmod(N (coTHH×(X))) = X × Σ X. Proof. This follows easily from the fact that 0 • 1 • N (coTHH×(X)) = X, N (coTHH×(X))=0 × X a • and N (coTHH×(X))= 0 for a ≥ 2. We next prove the special case of Theorem 3.15 where the module of cogenerators is one dimensional. Lemma 3.17. Let k be a field. Let X in dgmod≥0 be concentrated in a single nonnegative degree and be one dimensional in this degree. Then there is a quasi- isomorphism c Sc(X)|k coTHH⊗(S (X)) → Ω . Proof. First let the characteristic of k be 2 or X be concentrated in even degree, c so that S (X) =∼ Γk[x] for a generator x. Up to the internal degree induced by c X, we can use the results of Doi [11, 3.1] to compute coTHH⊗(S (X)) using a Sc(X) ⊗ Sc(X)-cofree resolution of Sc(X). Such a resolution is given by

△ f Sc(X) / Sc(X) ⊗ Sc(X) / Sc(X) ⊗ X ⊗ Sc(X) / 0, with f(xi ⊗xj )= xi−1 ⊗x⊗xj −xi ⊗x⊗xj−1, where x−1 = 0. Hence the homology c of coTHH⊗(S (X)) is concentrated in cosimplicial degrees 0 and 1. An explicit 0 • c ∼ c calculation gives that all elements in N (coTHH⊗(S (X))) = S (X) are cycles, 1 • c n n−i i while generating cycles in N (coTHH⊗(S (X))) are given by i=1 i · x ⊗ x P COMPUTATIONAL TOOLS FOR TOPOLOGICAL COHOCHSCHILD HOMOLOGY 15 for n ≥ 1. We identify xn in cosimplicial degree zero with xn ⊗ 1 ∈ U(Sc(X)) ⊗ −1 n n−i i n−1 −1 c −1 Λ(Σ X) and i=1 i · x ⊗ x with x ⊗ σ x ∈ U(S (X)) ⊗ Λ(Σ X). Now assume that X is concentrated in odd degree and that the characteristic of P c a c k is not 2, so that S (X)=Λk(x). Hence a typical element in coTHH (S (X)) is t t ta a • c of the form y = x 0 ⊗x 1 ⊗···⊗x with ti ∈{0, 1}. Now y ∈ N (coTHH (S (X))) ∗ • c if and only if t1 = ··· = ta = 1, and all differentials in N (coTHH (S (X))) are trivial. Identifying 1 ⊗ x⊗a with 1 ⊗ (σ−1x)a ∈ U(Sc(X)) ⊗ U(Sc(Σ−1X)) and x ⊗ x⊗a with x ⊗ (σ−1x)⊗a ∈ U(Sc(X)) ⊗ U(Sc(Σ−1X)) yields the result. The general case follows from the interplay of products and the cofree functor and the following result. Lemma 3.18 (Cf. [30, p. 719]). Let k be a field and let X be a graded k-module. Then Sc(X) =∼ colim Sc(V ), V ⊂X fin dim c c where the canonical projection πX : S (X) =∼ colimV ⊂Xfin dim S (V ) → X is given π by the colimit of the maps Sc(V ) −−→V V → X. Proof of Theorem 3.15. Recall from Example 2.12 that coTHH(C) of a counital coalgebra in dgmod≥0 can be computed as ∗ • coTHH⊗(C) = Totdgmod(N coTHH⊗(C)). Assume first that X has trivial differential. If X is finite dimensional, • c ∼ • c • c coTHH⊗(S (X)) = coTHH⊗(S (X1)) ⊗ ... ⊗ coTHH⊗(S (Xn)) for one dimensional k-vector spaces Xi. Hence Lemma 3.17 and the dual Eilenberg– Zilber map yield the desired quasi-isomorphism. Lemma 3.18 proves Theorem 3.15 for infinite dimensional cochain complexes X with trivial differential. If X is any nonnegatively graded cochain complex, applying the result for graded k-modules which we just proved shows that there is a morphism of double complexes ∗ • c Sc(X)|k N (coTHH⊗(S (X))) → Ω which is a quasi-isomorphism on each row, that is, if we fix the degree induced by the grading on X. Hence this induces a quasi-isomorphism on total complexes. Remark 3.19. Farinati and Solotar [13, Section 3] define a symmetric C-bicomodule 1 1 ΩC and a coderivation d: ΩC → C for any (ungraded) coaugmented coalgebra C 1 such that (ΩC , d) satisfies the following universal property: Every coderivation f : M → C from a symmetric C-bicomodule M to C factors as f = d ◦ f˜ for a unique C-bicomodule morphism f˜. Farinati and Solotar also give a construc- 1 tion of (ΩC , d) dual to the construction of the module of Kähler differentials asso- c ciated to a commutative algebra. To compare this with our definition of ΩS (X)|k, note that for X concentrated in degree zero 1 ∼ c ΩSc(X) = S (X) ⊗ X as a Sc(X)-bicomodule. Since Sc(X) is cofree, coderivations into Sc(X) correspond to maps into X, and the coderivation d: Sc(X)⊗X → Sc(X) is induced by the map ǫ⊗id: Sc(X)⊗X → X. If the characteristic of k is different from 2, this yields that 16 BOHMANN, GERHARDT, HØGENHAVEN, SHIPLEY, AND ZIEGENHAGEN

Sc(X)|k 1 1 Ω coincides with the exterior coalgebra ΛSc(X)(ΩSc(X)) on ΩSc(X) defined in [13, Section 6]. Remark 3.20. Analogous to Proposition [23, 5.4.6], the proof of Theorem 3.15 shows that the Theorem actually holds for any differential graded cocommutative couaug- mented coalgebra C whenever the underlying graded cocommutative coaugmented coalgebra is cofree. Remark 3.21. A similar result holds for unbounded chain complexes if we define Totdgmod and hence coTHH as a direct sum instead of a product in each degree. For nonnegatively graded cochain complexes both definitions of Totdgmod agree. However, if we use the homotopically correct definition of coTHH for unbounded chain complexes via the product total complex, Lemma 3.17 doesn’t hold for vector spaces X that are concentrated in degree −1.

4. A coBokstedt¨ Spectral Sequence While in algebraic cases we are able to understand certain good examples of coTHH by an analysis of the definition, in topological examples we require further tools. In this section, we construct a spectral sequence, which we call the coBökstedt spectral sequence. Recall that for a field k and a ring spectrum R, the skeletal filtration on the simplicial spectrum THH(R)• yields a spectral sequence 2 E∗,∗ = HH∗(H∗(R; k)) ⇒ H∗(THH(R); k), called the Bökstedt spectral sequence [1]. Analogously, for a coalgebra spectrum C, we consider the Bousfield–Kan spectral sequence arising from the cosimplicial spectrum coTHH•(C). We show in Theorem 4.6 that this is a spectral sequence of coalgebras. Since our spectral sequence is an instance of the Bousfield–Kan spectral sequence, its convergence is not immediate, but in the case where C is ∞ a suspension spectrum Σ+ X with X a simply connected space, we provide conditions in Corollary 4.5 below under which this spectral sequence converges to ∞ H∗(coTHH(Σ+ X); k). Let (Spec, ∧, S) denote a symmetric monoidal category of spectra, such as those given by [12], [22], or [27]. The notation ∧ means “smash product over S.” Let C be a coalgebra in this category with comultiplication △: C → C ∧ C and counit ε: C → S. Assume that C is cofibrant as a spectrum so that coTHH(C) has the correct homotopy type, as in Remark 2.4. Note that the spectral homology of C with coefficients in a field k is a graded k-coalgebra with structure maps

H∗(△) △: H∗(C; k) −−−−→ H∗(C ∧ C; k) =∼ H∗(C; k) ⊗k H∗(C; k),

H∗(ε) ε: H∗(C; k) −−−−→ k. Theorem 4.1. Let k be a field. Let C be a coalgebra spectrum that is cofibrant as a spectrum. The Bousfield–Kan spectral sequence for the cosimplicial spectrum • coTHH (C) gives a coBökstedt spectral sequence for calculating Ht−s(coTHH(C); k) with E2-page s,t k E2 = coHHs,t(H∗(C; k)). given by the classical coHochschild homology of H∗(C; k) as a graded k-module. COMPUTATIONAL TOOLS FOR TOPOLOGICAL COHOCHSCHILD HOMOLOGY 17

Proof. The spectral sequence arises as the Bousfield–Kan spectral sequence of a cosimplicial spectrum. We briefly recall the general construction. Let X• be a Reedy fibrant cosimplicial spectrum and recall that

n a Tot(X•)=eq (Xn)∆ ⇒ (Xb)∆ , n Y≥0 α∈∆([Ya],[b]) where ∆ is the cosimplicial space whose m-th level is the standard m-simplex ∆m. m Let skn∆ ⊂ ∆ denote the cosimplicial subspace whose m-th level is skn∆ , the n-skeleton of the m-simplex and set

m a • m skn∆ b skn∆ Totn(X ) := eq (X ) ⇒ (X ) . m Y≥0 α∈∆([Ya],[b]) The inclusions skn∆ ֒→ skn+1∆ induce the maps in a tower of ﬁbrations

• pn • • 0 ···→ Totn(X ) −→ Totn−1(X ) −→· · · → Tot0(X ) =∼ X .

in • pn • Let Fn −→ Totn(X ) −→ Totn−1(X ) denote the inclusion of the ﬁber and consider the associated exact couple: p • ∗ • π∗(Tot∗(X )) π∗(Tot∗(X ))

i∗ ∂ π∗(F∗).

This exact couple gives rise to a cohomologically graded spectral sequence {Er, dr} s,t s,t s+r,t+r−1 with E1 = πt−s(Fs) and differentials dr : Er → Er . It is a half plane spectral sequence with entering differentials. s s • The fiber Fs can be identified with Ω (N X ), where

(σ0,...,σs−1) s−1 s • s s−1 N X = eq X −−−−−−−−⇒ X ∗ i=0 Y We have isomorphisms s,t s s • ∼ s • ∼ s • E1 = πt−s(Ω (N X )) = πt(N X ) = N πt(X ) s • s+1 • and under these isomorphisms the differential d1 : N πt(X ) → N πt(X ) is i i identified with (−1) πt(δ ). Since the cohomology of the normalized complex agrees with the cohomology of the cosimplicial object, we conclude that the E2 term is P s,t ∼ s • i i E2 = H (πt(X ), (−1) πt(δ )). Let C be a coalgebra spectrum and considerP the spectral sequence arising from the Reedy fibrant replacement R(coTHH•(C) ∧ Hk). We have isomorphisms

n n ⊗k (n+1) π∗R(coTHH (C) ∧ Hk) =∼ π∗(coTHH (C) ∧ Hk) =∼ H∗(C; k) i The map π∗(δ ) corresponds to the i’th coHochschild diﬀerential under this identi- ﬁcation, therefore s,t k E2 = coHHs,t(H∗(C; k)). 18 BOHMANN, GERHARDT, HØGENHAVEN, SHIPLEY, AND ZIEGENHAGEN

From [4, IX 5.7], we have the following statement about the convergence of the coB¨okstedt spectral sequence. See also [16, VI.2] for a discussion of complete convergence. In [4, IX 5], the authors restrict to i ≥ 1 so that they are working with groups. Since we have abelian groups for all i, that restriction is not necessary here.

s,s+i s,s+i Proposition 4.2. If for each s there is an r such that Er = E∞ , then the coB¨okstedt spectral sequence for coTHH(C) converges completely to • π∗ Tot R(coTHH (C) ∧ Hk). There is a natural map P : Tot(RcoTHH•(C)) ∧ Hk → Tot R(coTHH•(C) ∧ Hk) arising from the natural construction of a map of the form Hom(X, Y ) ∧ Z → • Hom(X, Y ∧ Z). Since π∗(Tot(RcoTHH (C)) ∧ Hk) =∼ H∗(coTHH(C); k), we have the following.

s,s+i Corollary 4.3. If the conditions on Er in Proposition 4.2 hold and the map P described above induces an isomorphism in homotopy, then the coB¨okstedt spectral sequence for coTHH(C) converges completely to H∗(coTHH(C); k). Example 4.4. Let X be a simply connected space, let d: X → X × X denote the diagonal map and let c: X → ∗ be the map collapsing X to a point. We form a cosimplicial space X• with Xn = X×(n+1) and with cofaces

×i ×(n−i) ×(n+1) ×(n+2) X × d × X , 0 ≤ i ≤ n, δi : X → X , δi = ×n (τX,X×(n+1) (d × X ), i = n +1, and codegeneracies

×(n+2) ×(n+1) ×(i+1) ×(n−i) σi : X → X , σi = X × c × X for 0 ≤ i ≤ n. The cosimplicial space X• is the cosimplicial space coTHH•(X) of Deﬁnition 2.2 for the coalgebra (X,d,c) in the category of spaces; see Example 2.9. As noted earlier in Proposition 3.12, the cosimplicial space X• totalizes to the free loop space LX; see Example 4.2 of [3]. ∞ We can consider the spectrum Σ+ X as a coalgebra with comultiplication arising from the diagonal map d: X → X × X and counit arising from c: X →∗. We have an identiﬁcation of cosimplical spectra

• ∞ ∞ • coTHH (Σ+ X)=Σ+ (X ). The topological coHochschild homology is the totalization of a Reedy ﬁbrant replacement of the above cosimplicial spectrum. We have a natural map

∞ • ∞ • Σ+ Tot(X ) → Tot(Σ+ X ), as Malkiewich describes in Section 2 of [25], which after composing with the map to the totalization of a Reedy ﬁbrant replacement becomes a stable equivalence for simply connected X; see Proposition 2.22 of [25]. Hence we have a stable equivalence ∞ =∼ ∞ Σ+ LX −→ coTHH(Σ+ X). COMPUTATIONAL TOOLS FOR TOPOLOGICAL COHOCHSCHILD HOMOLOGY 19

Corollary 4.5. Let X be a simply connected space. If for each s there is an r such s,s+i s,s+i that Er = E∞ , the coB¨okstedt spectral sequence arising from the coalgebra ∞ Σ+ X converges completely to ∞ ∼ H∗(coTHH(Σ+ X); k) = H∗(LX; k). ∞ ∼ Proof. Since X is simply connected, by [25, 2.22] H∗(coTHH(Σ+ X); k) = H∗(LX; k). As in Corollary 4.3, we need to show that there is a weak equivalence

• ∞ • ∞ (1) Tot(RcoTHH (Σ+ X)) ∧ Hk ≃ TotR(coTHH (Σ+ X) ∧ Hk). First we consider the nth stage approximations:

• ∞ • ∞ Totn(RcoTHH (Σ+ X)) ∧ Hk ≃ TotnR(coTHH (Σ+ X) ∧ Hk).

The nth stages are weakly equivalent because the derived Totn functor is a finite homotopy limit functor. In spectra, this agrees with a finite homotopy colimit functor and hence the derived Totn commutes with smashing with Hk. Since these weak equivalences are compatible for all n, it follows that the inverse limits over n of both towers are weakly equivalent. On the right hand side, this inverse limit agrees with the right hand side of Equation 1. For the left hand sides to agree, we need to commute the inverse limit with − ∧ Hk. This is possible here because of the properties of this specific tower. In [25, 2.22], and in more detail in an earlier version [26, 4.6], Malkiewich • ∞ shows that the connectivity of the fibers of the tower {Totn R(coTHH (Σ+ X))} tend to infinity, and hence there is a pro-weak equivalence between this tower • ∞ and the constant tower given by {Tot R(coTHH (Σ+ X))}. That is, applying homotopy, πk, produces a pro-isomorphism of groups for each k. By [3, 8.5], it follows that there is a pro-isomorphism for each l between the constant tower • ∞ • ∞ {Hl Tot(RcoTHH (Σ+ X))} and the tower {Hl Totn R(coTHH (Σ+ X))}. It follows that there is a pro-weak equivalence

• ∞ ≃ • ∞ {Tot(RcoTHH (Σ+ X)) ∧ Hk} −→{Totn(RcoTHH (Σ+ X)) ∧ Hk}. By [4, III.3.1], it follows that the inverse limits are weakly equivalent. That is, there is a weak equivalence

• ∞ ≃ • ∞ Tot R(coTHH (Σ X)) ∧ Hk −→ lim(Totn R(coTHH (Σ X)) ∧ Hk). + n + This shows that − ∧ Hk commutes with inverse limits here as desired.

Our next result is to prove that the coBökstedt spectral sequence of Theorem 4.1 is a spectral sequence of coalgebras. We exploit this additional structure in Section 5 to make calculations of topological coHochschild homology in nice cases. We first describe a functor Hom(−, C): Setop → comCoalg, where C is a cocommutative coalgebra. In particular this will give us a cosimplicial spectrum 1 • Hom(S• , C) that agrees with the cosimplicial spectrum coTHH (C) defined in Def- inition 2.2. Let X be a set and C be a cocommutative coalgebra spectrum. Let Hom(X, C) be the indexed smash product C. x X ^∈ 20 BOHMANN, GERHARDT, HØGENHAVEN, SHIPLEY, AND ZIEGENHAGEN

If f : X → Y is a map of sets, we define a map of spectra Hom(Y, C) → Hom(X, C) as the product over Y of the component maps C → C −1 x∈f^ (y) given by iterated comultiplication over f −1(y). Here by convention the smash product indexed on the empty set is S, and the map to C → S is the counit map. Since C is cocommutative, this map doesn’t depend on a choosen ordering for applying the comultiplications and for each X, the comultiplication on C extends to define a cocommutative comultiplication on Hom(X, C). Hence Hom(−, C) is a op functor Set → comCoalg. If X∗ is a simplicial set, the composite Xop Hom(−,C) (∆op)op −−−→∗ Setop −−−−−−−→ comCoalg thus defines a cosimplicial cocommutative coalgebra spectrum. To obtain the cosimplicial coalgebra yielding coTHH(C), we take X∗ to be the 1 1 1 1 simplicial circle ∆•/∂∆•. Here ∆• is the simplicial set with ∆n = {x0,...,xn+1} where xt is the function [n] → [1] so that the preimage of 1 has order t. The face and degeneracy maps are given by:

xt t ≤ i δi(xt)= (xt−1 t>i

xt t ≤ i σi(xt)= (xt+1 t>i 1 1 The n-simplices of S∗ are obtained by identifying x0 and xn+1 so that Sn = 1 {x0,...,xn}. Comparing the coface and codegeneracy maps in Hom(S• , C) and coTHH•(C) shows that the two cosimplicial spectra are the same. Theorem 4.6. Let C be a connected cocommutative coalgebra that is cofibrant as a spectrum. Then the Bousfield–Kan spectral sequence described in Theorem 4.1 is a spectral sequence of k-coalgebras. In particular, for each r> 1 there is a coproduct ∗∗ ∗∗ ∗∗ ψ : Er → Er ⊗k Er , and the differentials dr respect the coproduct. Proof. Below we will construct the coproduct using natural maps of spectral sequences ∗∗ ∇ ∗∗ AW ∗∗ ∗∗ Er −−→ Dr ←−−−− Er ⊗k Er where the map AW is an isomorphism for r ≥ 2. Recall that the cosimplicial spectrum coTHH•(C) can be identified with the 1 ∗∗ cosimplicial spectrum Hom(S• , C). Let Dr denote the Bousfield–Kan spectral 1 1 sequence for the cosimplicial spectrum Hom((S ∐ S )•, C) ∧ Hk. The codiago- 1 1 1 1 nal map ∇: S ∐ S → S is a simplicial map and gives a map Hom(S• , C) → 1 1 ∗∗ ∗∗ Hom((S ∐ S )•, C). The map ∇: Er → Dr is induced by this codiagonal map. Let X• denote the cosimplicial Hk-coalgebra spectrum coTHH(C)• ∧ Hk. Note 1 1 that the cosimplicial spectrum Hom((S ∐ S )•, C) ∧ Hk is the cosimplicial spec- n n trum [n] → X ∧Hk X . This is the diagonal cosimplicial spectrum associated p q to a bicosimplicial spectrum, (p, q) → X ∧Hk X . We will denote this diagonal COMPUTATIONAL TOOLS FOR TOPOLOGICAL COHOCHSCHILD HOMOLOGY 21

• • ∗∗ cosimplicial spectrum by diag(X ∧Hk X ). The spectral sequence Dr is given by • • the Tot tower for the Reedy fibrant replacement R(diag(X ∧Hk X )). Note that our cosimplicial spectra are in fact cosimplicial Hk-modules. We can apply the standard equivalence between Hk-modules and chain complexes of k- modules [33]. By hypothesis, our Hk-modules are connective and therefore can be replaced by non-negatively graded chain complexes of k-modules. By the Dold–Kan correspondence, non-negatively graded chain complexes of k-modules are equivalent to simplicial k-modules. This also holds on the level of diagram categories and therefore our cosimplicial Hk-module spectra can be identified as cosimplicial simplicial k-modules. This string of equivalences is weakly monoidal in the sense of [32]. The Bousfield–Kan results in the simplicial setting [5, 6] thus apply, giving us a map of spectral sequences ∗∗ ∗∗ ∗∗ Er ⊗k Er → Dr . By Bousfield and Kan, on the r = 1 page, this map is given by the Alexander– Whitney map: ∗∗ ∗∗ ∗ • ∗ • ∗ • • ∗∗ E1 ⊗k E1 = N (π∗X ) ⊗k N (π∗X ) → N (π∗X ⊗k π∗X )= D1 . By the cosimplicial analog of the Eilenberg–Zilber theorem (see, for example, [15]) this induces an isomorphism on cohomology

∼ ∗ ∗,∗ ∗ ∗,∗ ∼ ∗ ∗,∗ ∗,∗ = / ∗ ∗,∗ ∼ ∗∗ H (E1 ) ⊗k H (E1 ) = H (E1 ⊗k E1 ) H (D1 ) = D2 , where the first isomorphism is given by the Künneth theorem. This composite is the ∼ ∗,∗ ∗,∗ = ∗∗ map E2 ⊗k E2 −→ D2 . By induction and repeated application of the Künneth ∼ ∗,∗ ∗,∗ = ∗∗ theorem, we get equivalences Er ⊗k Er −→ Dr for all r > 1. The composite of ∇ with the inverse of this isomorphism gives us the desired coproduct.

5. Computational results Now that we have developed the coBökstedt spectral sequence and its coalgebra structure, we use these structures to make computations of the homology of coTHH(C) for certain coalgebra spectra C. We also prove that in certain cases this spectral sequence collapses. We first make some elementary computations of coHochschild homology, which we will later use as input to the coHH to coTHH spectral sequence described in Theorem 4.1. The coalgebras we consider are the underlying coalgebras of Hopf algebras, specifically, of exterior Hopf algebras, polynomial Hopf algebras and divided power Hopf algebras. As mentioned in Example 3.8, the exterior Hopf algebra and the divided power Hopf algebras give the free cocommutative coalgebras on graded k-modules concentrated in odd degree or even degree, respectively. Recall from Ex- ample 3.8 that Λk(y1,y2,... ) denotes the exterior Hopf algebra on generators yi in odd degrees and that Γk[x1, x2,... ] denotes the divided power Hopf algebra on generators xi in even degrees. Let k[w1, w2,... ] denote the polynomial Hopf algebra on j j k j−k generators wi in even degree. The coproduct is given by △(wi )= k k wi ⊗wi . In the following computations, we assume that all the Hopf algebras in question only have finitely many generators in any given degree. We workP over a field k. The following computation of coHochschild homology is the main result we use as input for our spectral sequence. 22 BOHMANN, GERHARDT, HØGENHAVEN, SHIPLEY, AND ZIEGENHAGEN

Proposition 5.1. Let the cogenerators xi be of even nonnegative degree and let the cogenerators yi be of odd nonnegative degree. We evaluate the coHochschild homology as a coalgebra for the following cases of free cocommutative coalgebras:

coHH∗(Γk[x1, x2,... ]) = Γk[x1, x2,... ] ⊗ Λk(z1,z2,... ), where deg(zi)=(1, deg(xi)).

coHH∗(Λk(y1,y2,... ))=Λk(y1,y2,... ) ⊗ k[w1, w2,... ] where deg(wi)=(1, deg(yi)). Before proving the proposition, we recall some basic homological coalgebra. Con- sider any coalgebra C over a ﬁeld k. Let M be a right C-comodule and N be a left C-comodule with corresponding maps ρM : M → M ⊗ C and ρN : N → C ⊗ N. Recall that the cotensor product of M and N over C, MCN, is deﬁned as the kernel of the map

ρM ⊗ 1 − 1 ⊗ ρN : M ⊗ N → M ⊗ C ⊗ N. The Cotor functors are the derived functors of the cotensor product. In other n n words, CotorC (M,N)= H (MCX) where X is an injective resolution of N as a left C-comodule. Let Ce = C ⊗ Cop. Let N be a (C, C)-bicomodule. Then N can be regarded as e n a right C -comodule. As shown in Doi [11], coHHn(N, C) = CotorCe (N, C).

Lemma 5.2. For C =Γk[x1, x2,...] or C =Λk(y1,y2,...),

coHH(C)= C ⊗ CotorC (k, k). n Proof. As above, coHHn(C) = CotorCe (C, C). Observe that C is a Hopf algebra. Let S denote the antipode in C. As in Doi [11, Section 3.3], we consider the coalgebra map ∇: Ce → C op given by ∇(c ⊗ d ) = cS(d). Given a (C, C)-bicomodule M, let M∇ denote M viewed as a right C-comodule via the map ∇. Let k → Y 0 → Y 1 → · · · be an injective resolution of k as a left C-comodule. e e n When C is a Hopf algebra, (C )∇ is free as a right C-comodule, and (C )∇C Y is injective as a left Ce-comodule [11]. The sequence e e 0 e 1 (C )∇Ck → (C )∇C Y → (C )∇CY → · · · e e is therefore an injective resolution of (C )∇C k =∼ C as a left C -comodule. Coten- soring over Ce with C we have: e 0 e 1 CCe ((C )∇CY ) → CCe ((C )∇CY ) → · · · which gives: 0 1 C∇C Y → C∇CY → · · · ∼ n ∼ n Therefore coHHn(C) = CotorCe (C, C) = CotorC (C∇, k). In our cases C is cocom- n ∼ mutative, so C∇ has the trivial C-comodule structure. Therefore CotorC (C∇, k) = n C ⊗ CotorC (k, k).

We now want to compute CotorC (k, k) for the coalgebras C that we are interested in. COMPUTATIONAL TOOLS FOR TOPOLOGICAL COHOCHSCHILD HOMOLOGY 23

Proposition 5.3. For C = Γk[x1, x2,...], CotorC (k, k)=Λk(z1,z2,...) where deg(zi)=(1, deg(xi)). For C =Λk(y1,y2,...), CotorC (k, k)= k[w1, w2,...] where deg(wi)=(1, deg(yi)). Proof. Note that in both cases C is a cocommutative coalgebra that in each ﬁxed degree is a free and ﬁnitely generated k-module. Let A denote the Hopf algebra dual of C. As in [31] we conclude that CotorC(k, k) is a Hopf algebra, which is dual to the Hopf algebra TorA(k, k). Recall that the Hopf algebra dual of Γk[x1, x2,... ] is k[x1, x2,... ] and the Hopf algebra Λk(y1,y2,... ) is self dual. We recall the classical results:

k[x1,x2,...] Tor (k, k) =∼ Λk(z1,z2,...),

Λk (y1,y2,...) Tor (k, k) =∼ Γk[w1, w2,...] where the degree of zi is (1, deg(xi)) and the degree of wi is (1, deg(yi)). The result follows.

We have now proven Proposition 5.1. These results provide the input for the coB¨okstedt spectral sequence. They are the starting point for the following theorem, which says that this spectral sequence collapses at E2 in the case where the homology of the input is cofree on cogenerators of even nonnegative degree. Theorem 5.4. Let C be a cocommutative coassociative coalgebra spectrum that is coﬁbrant as a spectrum, and whose homology coalgebra is

H∗(C; k)=Γk[x1, x2,... ], where the xi are cogenerators in nonnegative even degrees and there are only ﬁnitely many cogenerators in each degree. Then the coB¨okstedt spectral sequence for C collapses at E2, and

E2 =∼ E∞ =∼ Γk[x1, x2,... ] ⊗ Λk(z1,z2,... ), with xi in degree (0, deg(xi)) and zi in degree (1, deg(xi)). Proof. By Proposition 5.1,

coHH∗(Γk[x1, x2,... ]) = Γk[x1, x2,... ] ⊗ Λk(z1,z2,... ). A conilpotent coalgebra C in graded k-modules is called cogenerated by Y if there is a surjection π : C → Y such that

(△n) T c(π) C −−−−→n T c(C) −−−−→ T c(Y ) is injective. Every cofree cocommutative coalgebra Sc(X) is cogenerated by X under the canonical projection π : Sc(X) → X, since the composite

(∆n) T c(π) Sc(X) −−−−→n T c(Sc(X)) −−−−→ T c(X) is the inclusion of Sc(X) into T c(X). Let p be the characteristic of k. If p 6= 2, coHH∗(Γk[x1, x2,... ]) is the cofree cocommutative conilpotent coalgebra cogenerated by {x1, x2,...,z1,z2,... } with respect to the total grading. If p = 2, it is easy to verify that Γk[x1, x2,... ] ⊗ Λk(z1,z2,... ) is cogenerated by {x1, x2,...,z1,z2,... } as well. 24 BOHMANN, GERHARDT, HØGENHAVEN, SHIPLEY, AND ZIEGENHAGEN

If C is cogenerated by π : C → Y , any coderivation d: C → C is completely determined by πd: Since n+1 △nd = C⊗i−1 ⊗ d ⊗ C⊗n+1−i ◦△n i=1 X and hence n+1 c n ⊗i−1 ⊗n+1−i n T (π)(△ )nd = ( π ⊗ πd ⊗ π △ )n, i=1 c n X the injectivity of T (π)(△ )n yields that πd determines d. In particular, if πd = 0, this implies d = 0. We now return to the coBökstedt spectral sequence. Assume that for some r ≥ 2 2 r−1 we already know that d ,...,d vanish and thus the pages E2 =∼ ··· =∼ Er−1 are as claimed. The differential dr has bidegree (r, r − 1). Since the cogenerators are s,t of bidegrees (0,t) and (1,t) and Er−1 = 0 if s < 0, the cogenerators cannot be in the image of dr and we find that πdr = 0 if r ≥ 2.

Recall that for a space X the suspension spectrum Σ∞X is an S-coalgebra. This provides a wealth of examples for which we can use the computational tools of Theorem 5.4. Before stating particular results, we discuss the motivation for studying topological coHochschild homology of suspension spectra. Recall the following theorem from Malkiewich [25] and Hess–Shipley [18]: Theorem 5.5. For X a simply connected space, ∞ ∞ THH(Σ+ (ΩX)) ≃ coTHH(Σ+ X) ∞ Consequently, for a simply connected space X, to understand THH of Σ+ (ΩX), one can instead compute the coTHH of the suspension spectrum of X. In some cases this topological coHochschild homology computation is more accessible. In ∞ particular, to compute the homology H∗(THH(Σ+ (ΩX)); k) using the Bökstedt spectral sequence, the necessary input is the Hochschild homology of the algebra ∞ ∞ H∗(Σ+ (ΩX); k). To compute H∗(coTHH(Σ+ X); k) the spectral sequence input ∞ is the coHochschild homology of the coalgebra H∗(Σ+ X; k). In some cases this homology coalgebra is easier to work with than the homology algebra of the loop space, making the topological coHochschild homology calculation a more accessible path to studying the topological Hochschild homology of based loop spaces. For instance, the Lie group calculations in Example 5.7 are more approachable via the coTHH spectral sequence. Topological Hochschild homology of based loop spaces is of interest due to connections to algebraic K-theory, free loop spaces, and string topology, which we will now recall. ∞ ∞ Recall Bökstedt and Waldhausen showed that THH(Σ+ (ΩX)) ≃ Σ+ LX, where LX is the free loop space, LX = Map(S1,X). The homology of a free loop space, H∗(LX), is the main object of study in the field of string topology [8, 10]. Since ∞ ∞ coTHH(Σ+ X) ≃ Σ+ LX for simply connected X, topological coHochschild homology gives a new way of approaching this homology. Topological coHochschild homology of suspension spectra also has connections to algebraic K-theory. Waldhausen’s A-theory of spaces A(X) is equivalent to COMPUTATIONAL TOOLS FOR TOPOLOGICAL COHOCHSCHILD HOMOLOGY 25

∞ K(Σ+ ΩX). Recall that there is a trace map from algebraic K-theory to topological Hochschild homology ∞ ∞ A(X) ≃ K(Σ+ ΩX) → THH(Σ+ ΩX). When X is simply connected this gives us a trace map to topological coHochschild homology as well: ∞ ∞ ∞ A(X) ≃ K(Σ+ ΩX) → THH(Σ+ ΩX) ≃ coTHH(Σ+ X). In the remainder of this section we will make some explicit computations of the homology of coTHH of suspension spectra. Example 5.6. Let X be a product of n copies of CP ∞. As a coalgebra ∞ ∼ H∗(Σ+ X; k) = Γk(x1, x2,...xn) where deg(xi) = 2. By Theorem 5.4, there is an isomorphism of graded k-modules ∞ ∼ H∗(coTHH(Σ+ X); k) = Γk(x1, x2,...xn) ⊗ Λk(ˆx1, xˆ2,... xˆn). Note that any space having cohomology with polynomial generators in even degrees will give us a suspension spectrum whose homology satisﬁes the conditions of Theorem 5.4. Hence this theorem allows us to compute the topological coHochschild homology of various suspension spectra, as in the example below. Example 5.7. Theorem 5.4 directly yields the following computations of topological coHochschild homology. The isomorphisms are of graded k-modules. ∞ ∼ H∗(coTHH(Σ+ BU(n)); k) = Γk(y1,...yn) ⊗ Λk(ˆy1,... yˆn)

|yi| =2i, |yˆi| =2i − 1

∞ ∼ H∗(coTHH(Σ+ BSU(n)); k) = Γk(y2,...yn) ⊗ Λk(ˆy2,... yˆn)

|yi| =2i, |yˆi| =2i − 1

∞ ∼ H∗(coTHH(Σ+ BSp(n)); k) = Γk(z1,...zn) ⊗ Λk(ˆz1,... zˆn)

|zi| =4i, |zˆi| =4i − 1 With some restrictions on the prime p, we also immediately get the mod p homology of the topological coHochschild homology of the suspension spectra of the classifying spaces BSO(2k), BG2, BF4, BE6, BE7, and BE8. Note that these computations also yield the homology of the free loop spaces H∗(LBG; Fp) and the ho- ∞ F mology H∗(THH(Σ+ G); p), for G = U(n),SU(n),Sp(n), SO(2k), G2, F4.E6, E7, and E8. The topological coHochschild homology calculations also follow directly for products of copies of any of these classifying spaces BG.

Appendix A. Proof of Lemma 2.6 We prove the assertion of Lemma 2.6. This shows that coTHH•(C) is Reedy ﬁbrant when C is a counital coaugmented coalgebras in the category of graded k-modules for a commutative ring k. Proof of Lemma 2.6. Let η : k → C be the coaugmentation or the k-module map ⊗n ⊗n+1 such that ǫη = id. Consider the maps ηi : C → C given by applying η between the i + 1 and i +2 copy of C; that is ⊗i+1 ⊗n−i−1 ηi = id ⊗ η ⊗ id . 26 BOHMANN, GERHARDT, HØGENHAVEN, SHIPLEY, AND ZIEGENHAGEN

The indexing has been chosen to match the indexing on the codegeneracy maps σi in the sense that σiηi = id. Direct calculation yields the following additional relations among the ηi maps and codegeneracies:

ηj ηi = ηiηj−1 i < j σj ηi = ηiσj−1 i < j σiηj = ηj−1σi i < j

 The last two relations in particular imply that σi+1ηi = ηiσi = σiηi+1. To get a sense for these operations, note that if c0 ⊗···⊗ cn is a simple tensor, then

ηiσi(c0 ⊗···⊗ cn)= c0 ⊗···⊗ ci ⊗ ηǫ(ci+1) ⊗ ci+1 ⊗···⊗ cn so that ηiσi replaces the i + 1th tensor factor ci+1 with the result of sending it to k via the counit and then mapping back up via the coaugmentation. • From Hirschhorn, we know that the matching space Mn(X ) for a cosimplicial graded k-module X• is given by

• n−1 ×n Mn(X )= {(x0,...,xn−1) ∈ (X ) | σi(xj )= σj−1(xi) for 0 ≤ i < j ≤ n − 1}.

• • • Let X = coTHH(C) and consider (x0,...,xn−1) ∈ Mn(X ). We deﬁne y ∈ coTHH(C)n = C⊗n+1 to be the following sum:

n−1 i j y = ηi xi + (−1) ηl σl ··· ηl σl (xi) .  1 1 j j  i=0 j=1 l <

That is, for each i, we apply ηi to xi plus a signed sum over all tuples of distinct numbers between 0 and i−1 of applying ηǫ to xi in the spots corresponding to these numbers. The is sign determined by the number of elements in a tuple. (In fact, we can think of xi itself as corresponding to the “empty tuple” and so unify our description of these terms, but that seems more obfuscating than strictly necessary.) We show that σs(y)= xs for all 0 ≤ s ≤ n−1, so that the matching map applied to y yields (x0,...,xn−1). First, break σs(y) into terms corresponding to is.

s−1 i j σs(y)= σsηi xi + (−1) ηl σl ··· ηl σl (xi)  1 1 j j  i=0 j=1 l <

s−1 i j σs(y)= ηiσs xi + (−1) ηl σl ··· ηl σl (xi) −1  1 1 j j  i=0 j=1 l < s all vanish and that the terms withi ≤ s cancel to leave only xs. Observe that whenever l

(2) σsηlσl = ηlσs−1σl = ηlσlσs.

That is, σs “commutes with” the operation ηlσl for l

σsηl1 σl1 ··· ηlj σlj = ηl1 σl1 ··· ηla σla σsηla+1 σla+1 ··· ηlj σlj We then split the operation in the summation term of Equation 3 into two cases: the summation over tuples where s = la+1 and the summation over tuples where s

If la+1 = s, then σsηla+1 σla+1 = σsηsσs = σs, so these tuples yield the operation i j −σs + (−1) ηl1 σl1 ··· ηla σla σsηla+2 σla+2 ··· ηlj σlj j=2 j-tuples X containingX s

The tuples where la+1 >s yield the operation i j (−1) ηl1 σl1 ··· ηla σla σsηla+1 σla+1 ··· ηlj σlj . j=1 j-tuples not X containingX s

When we sum these two operations, all the terms cancel except −σs because every j-tuple containing s yields a j − 1-tuple not containing s after omitting the s. Thus i j σs( (−1) ηl1 σl1 ··· ηlj σlj (xi)) = −σs(xi) j=1 l < s vanishes. 28 BOHMANN, GERHARDT, HØGENHAVEN, SHIPLEY, AND ZIEGENHAGEN

Our next task is to understand the terms of the form i ηiσs xi + ηl σl ··· ηl σl (xi) −1  j 1 j j  j=1 l <

ηiσs−1(xi)+ ηiηlj σl1 ··· ηlj σlj σs−1(xi) j=1 l <

ηiσs−1(xi)+ ηlj σl1 ··· ηlj σlj ηiσs−1(xi) j=1 l <

+ xs + ηl1 σl1 ··· ηlj σlj (xs) j=2 l <

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Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN, 37240, USA E-mail address: [email protected]

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